A new self-consistent closure is formulated for the additive hard sphere mixtures at high densities (eta =0.49) within the Ornstein-Zernike integral equation approach. Diameter ratios (sigma (SS)/sigma (BB)) from 0.3 to 0.9 and several compositions are examined. The consistencies include the thermodynamic ones (e.g., pressure consistency, and Gibbs-Duhem relation), and the structural ones (e.g., the zero-separation theorems). The bridge functions have built-in "flexibility" that can be adapted to the consistency requirements. Comparison with Monte Carlo simulation shows that the present closure yields highly accurate results. The contact values and zero-separation values are more accurate than those obtained from the conventional closures, such as the Percus-Yevick and Martynov-Sarkisov closures. A structural theory for hard sphere mixtures has been formulated that is accurate and consistent at the same time.